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Question Number 13942 by FilupS last updated on 25/May/17

$$\mathrm{for}\mathit{v}=\left[\begin{array}{c}y\\ x\end{array}\right],\mathit{v}\in {\mathbb{R}}^2$$$$\mathit{v}\mathrm{has}\mathrm{basis}\mathrm{vectors}\hat{i}\mathrm{and}\hat{j}$$$$$$$$\mathrm{Assume}\mathrm{we}\mathrm{apply}\mathrm{a}\mathrm{basis}\mathrm{transform}\mathrm{to}$$$$\mathrm{obtain}\mathrm{new}\mathrm{basis}\mathrm{vectors}{\hat{i}}^{\prime }\mathrm{and}{\hat{j}}^{\prime }$$$$$$$$\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{new}{\mathit{v}}^{\prime }?$$

Answered by ajfour last updated on 25/May/17

$${\mathit{v}}^{\prime }=\left[\begin{array}{c}{y}^{\prime }\\ x^{\prime }\end{array}\right]=\left[\begin{array}{cc}\hat{i}.{\hat{j}}^{\prime }& \hat{j}.{\hat{j}}^{\prime }\\ \hat{i}.{\hat{i}}^{\prime }& \hat{j}.{\hat{i}}^{\prime }\end{array}\right]\left[\begin{array}{c}y\\ x\end{array}\right]$$$$(maybe..).$$$$$$

Commented by FilupS last updated on 26/May/17

$$\left[\begin{array}{cc}\hat{i}.{\hat{j}}^{\prime }& \hat{j}.{\hat{j}}^{\prime }\\ \hat{i}.{\hat{i}}^{\prime }& \hat{j}.{\hat{i}}^{\prime }\end{array}\right]\mathrm{I}\mathrm{have}\mathrm{a}\mathrm{feeling}\mathrm{this}\mathrm{is}\mathrm{incorrect}$$$$$$$$\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{remember},\mathrm{but}\mathrm{I}\mathrm{think}\mathrm{it}\mathrm{should}$$$$\mathrm{be}\mathrm{something}\mathrm{like}\left[\begin{array}{cc}\hat{i}& {\hat{i}}^{\prime }\\ \hat{j}& {\hat{j}}^{\prime }\end{array}\right]$$

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